St Andrew’s Academy
Mathematics Department
Higher Mathematics
EXPONENTIALS &
LOGARITHMS
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Exponentials and Logarithms Contents
Exponentials and Logarithms 1
1 Exponentials EF 1 2 Logarithms EF 3 3 Laws of Logarithms EF 3 4 Exponentials and Logarithms to the Base e EF 6 5 Exponential and Logarithmic Equations EF 7 6 Graphing with Logarithmic Axes EF 10 7 Graph Transformations EF 14
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Exponentials and Logarithms
1 Exponentials EF We have already met exponential functions in the notes on Functions and Graphs..
A function of the form ( ) xf x a= , where 0a > is a constant, is known as an exponential function to the base a.
If 1a > then the graph looks like this:
This is sometimes called a growth function.
If 0 1a< < then the graph looks like this:
This is sometimes called a decay function.
Remember that the graph of an exponential function ( ) xf x a= always passes through ( ) 0,1 and ( ) 1, a since:
( ) 00 1f a= = , ( ) 11f a a= = .
O
1 ( ) 1, a
y
x
, 0 1xy a a= < <
O
1 ( ) 1, a
y
x
, 1xy a a= >
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EXAMPLES
1. The otter population on an island increases by 16% per year. How many full years will it take the population to double? Let 0u be the initial population.
( )( )
1 0
22 1 0 0
2 33 2 0 0
0
1·16 (116% as a decimal)
1·16 1·16 1·16 1·16
1·16 1·16 1·16 1·16 1·16 .n
n
u u
u u u u
u u u u
u u
=
= = =
= = =
=
For the population to double after n years, we require 02nu u≥ .
We want to know the smallest n which gives 1·16n a value of 2 or more, since this will make nu at least twice as big as 0u .
Try values of n until this is satisfied.
2
3
4
5
If 2, 1·16 1·35 2If 3, 1·16 1·56 2If 4, 1·16 1·81 2If 5, 1·16 2·10 2
nnnn
= = <= = <= = <= = >
On a calculator: 1 ⋅ 1 6 = 1 ⋅ 1 6 ANS = =
Therefore after 5 years the population will double.
2. The efficiency of a machine decreases by 5% each year. When the efficiency drops below 75%, the machine needs to be serviced. After how many years will the machine need to be serviced? Let 0u be the initial efficiency.
( )( )
1 02
2 1 0 02 3
3 2 0 0
0
0·95 (95% as a decimal)0·95 0·95 0·95 0·950·95 0·95 0·95 0·95
0·95 .nn
u uu u u uu u u u
u u
== = == = =
=
When the efficiency drops below 00·75u (75% of the initial value) the machine must be serviced. So the machine needs serviced after n years if 0·95 0·75n ≤ .
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Try values of n until this is satisfied:
2
3
4
5
6
If 2, 0·95 0·903 0·75If 3, 0·95 0·857 0·75If 4, 0·95 0·815 0·75If 5, 0·95 0·774 0·75If 6, 0·95 0·735 0·75
nnnnn
= = >= = >= = >= = >= = <
Therefore after 6 years, the machine will have to be serviced.
2 Logarithms EF Having previously defined what a logarithm is (see the notes on Functions and Graphs) we now look in more detail at the properties of these functions.
The relationship between logarithms and exponentials is expressed as:
log yay x x a= ⇔ = where , 0a x > .
Here, y is the power of a which gives x. EXAMPLES
1. Write 35 125= in logarithmic form. 3
55 125 3 log 125= ⇔ = .
2. Evaluate 4log 16 .
The power of 4 which gives 16 is 2, so 4log 16 2= .
3 Laws of Logarithms EF There are three laws of logarithms which you must know.
Rule 1
( )log log loga a ax y xy+ = where , , 0a x y > .
If two logarithmic terms with the same base number (a above) are being added together, then the terms can be combined by multiplying the arguments (x and y above).
EXAMPLE
1. Simplify 5 5log 2 log 4+ .
( )5 5
5
5
log 2 log 4log 2 4log 8.
+= ×=
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Rule 2
( )log log loga a axyx y− = where , , 0a x y > .
If a logarithmic term is being subtracted from another logarithmic term with the same base number (a above), then the terms can be combined by dividing the arguments (x and y in this case).
Note that the argument which is being taken away (y above) appears on the bottom of the fraction when the two terms are combined.
EXAMPLE
2. Evaluate 4 4log 6 log 3− .
( )12
4 4
4
4
63
12
log 6 log 3
loglog 2
(since 4 4 2).
−
==
= = =
Rule 3
log logna ax n x= where , 0a x > .
The power of the argument (n above) can come to the front of the term as a multiplier, and vice-versa.
EXAMPLE
3. Express 72 log 3 in the form 7log a .
72
7
7
2 log 3log 3log 9.
==
Squash, Split and Fly
You may find the following names are a simpler way to remember the laws of logarithms.
• ( )log log loga a ax y xy+ = – the arguments are squashed together by multiplying.
• ( )log log loga a axyx y− = – the arguments are split into a fraction.
• log logna ax n x= – the power of an argument can fly to the front of the
log term and vice-versa.
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Note
When working with logarithms, you should remember:
log 1 0a = since 0 1a = , log 1a a = since 1a a= .
EXAMPLE
4. Evaluate 7 3log 7 log 3+ .
7 3log 7 log 31 12.
+= +=
Combining several log terms
When adding and subtracting several log terms in the form loga b , there is a simple way to combine all the terms in one step.
Multiply the arguments of the positive log terms in the numerator. Multiply the arguments of the negative log terms in the denominator.
EXAMPLES
5. Evaluate 12 12 12log 10 log 6 log 5+ − .
12 12 12
12
12
log 10 log 6 log 510 6log
5log 121.
+ −
× =
==
6. Evaluate 6 6log 4 2 log 3+ .
( )
6 6
26 6
6 6
6
6
2
log 4 2 log 3
log 4 log 3log 4 log 9log 4 9log 36
2 (since 6 36).
+
= += +
= ×=
= =
OR
( )( )( )
6 6
26 6
6 6
6 6
6
6
6
log 4 2 log 3
log 2 2 log 32 log 2 2 log 3
2 log 2 log 3
2 log 2 32 log 62 (since log 6 1).
+
= += +
= +
= ×
== =
( )12 log
12 12log log+ +10 6
12log− 5
( ) loga
arguments of positive log terms
arguments of negative log terms
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4 Exponentials and Logarithms to the Base e EF The constant e is an important number in Mathematics, and occurs frequently in models of real-life situations. Its value is roughly 2⋅718281828 (to 9 d.p.), and is defined as:
( )11 as n
e nn= + →∞ .
If you try very large values of n on your calculator, you will get close to the value of e. Like π , e is an irrational number.
Throughout this section, we will use e in expressions of the form: • xe , which is called an exponential to the base e; • loge x , which is called a logarithm to the base e. This is also known as the
natural logarithm of x, and is often written as ln x (i.e. ≡ln logex x ).
EXAMPLES
1. Calculate the value of log 8e . .log 8 2 08 (to 2 d.p.).e =
2. Solve log 9e x = .
9
log 9
so 8103·08 (to 2 d.p.).
e x
x ex
=
==
3. Simplify ( ) ( )4 log 2 3log 3e ee e− expressing your answer in the form log loge ea b c+ − where a, b and c are whole numbers.
( ) ( )
4 3
4 log 2 3log 34 log 2 4 log 3log 3 3log4 log 2 4 3log 3 31 4 log 2 3log 3
1 log 2 log 31 log 16 log 27.
e e
e e e e
e e
e e
e e
e e
e ee e
−= + − −= + − −= + −
= + −= + −
OR ( ) ( )( ) ( )( )( )
4 3
4
3
4
3
4 log 2 3log 3
log 2 log 3
2log
3
16log2716log27
log log 16 log 271 log 16 log 27.
e e
e e
e
e
e
e e e
e e
e e
e e
ee
eee
e
−
= −
=
= =
= + −= + −
On a calculator: ex 9 =
On a calculator: ln 8 =
Remember
( ) =n n nab a b .
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5 Exponential and Logarithmic Equations EF Many mathematical models of real-life situations use exponentials and logarithms. It is important to become familiar with using the laws of logarithms to help solve equations.
EXAMPLES
1. Solve log 13 log log 273a a ax+ = for 0x > .
log 13 log log 273log 13 log 273
13 273 (since log log )21.
a a a
a a
a a
xxx x y x yx
+ === = ⇔ ==
2. Solve ( ) ( )11 11log 4 3 log 2 3 1x x+ − − = for 32x > .
( ) ( )
( )
11 11
11
1
log 4 3 log 2 3 14 3log 12 34 3 11 11 (since log )2 34 3 11 2 34 3 22 33
18 362.
ay
x x
xxx x y x axx xx x
xx
+ − − =
+ = − +
= = = ⇔ =−+ = −+ = −
==
3. Solve ( ) ( ) ( )log 2 1 log 3 10 log 11a a ap p p+ + − = for 4p > .
( ) ( ) ( )( )( ) ( )( )( )
( )( )
2
2
log 2 1 log 3 10 log 11
log 2 1 3 10 log 11
2 1 3 10 11
6 20 3 10 11 0
6 28 10 0
3 1 5 0
a a a
a a
p p p
p p p
p p p
p p p p
p p
p p
+ + − =
+ − = + − =
− + − − =
− − =
+ − =
13
3 1 0p
p
+ =
= −
or 5 05.
pp
− ==
Since we require 4p > , 5p = is the solution.
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Dealing with Constants
Sometimes it may help to write constants as logs to solve equations. EXAMPLE
4. Solve 2 2log 7 log 3x= + for 0x > .
Write 3 in logarithmic form:
2 2
32
2
3 3 13log 2 (since log 2 1)
log 2log 8.
= ×= =
==
Use this in the equation:
2 2 2
2 2
78
log 7 log log 8log 7 log 8
7 8
.
xx
x
x
= +==
=
OR
( )
2 2
2 2
27
log 7 log 3log 7 log 3
log 3.x
xx= +
− =
=
Converting from log to exponential form:
3
37
7278
2
.
x
x
=
=
=
Solving Equations with unknown Exponents
If an unknown value (e.g. x) is the power of a term (e.g. xe or 10x ), and its value is to be calculated, then we must take logs on both sides of the equation to allow it to be solved.
The same solution will be reached using any base, but calculators can be used for evaluating logs to the base e and 10.
EXAMPLES
5. Solve 7xe = .
Taking loge of both sides: OR Taking 10log of both sides:
log log 7log log 7 ( log 1)
log 71·946 (to 3 d.p.).
xe e
e e e
e
ex e e
xx
== ===
10 10
10 10
10
10
log log 7log log 7
log 7log
1·946 (to 3 d.p.).
xex e
xe
x
==
=
=
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6. Solve 3 15 40x+ = .
( )
3 1log 5 log 403 1 log 5 log 40
log 403 1log 5
3 1 2·29203 1·2920
0·431 (to 3 d.p.).
xe e
e e
e
e
x
x
xxx
+ =
+ =
+ =
+ ===
Exponential Growth and Decay
Recall from Section 1 that exponential functions are sometimes known as growth or decay functions. These often occur in models of real-life situations.
For instance, radioactive decay can be modelled using an exponential function. An important measurement is the half-life of a radioactive substance, which is the time taken for the mass of the radioactive substance to halve.
EXAMPLE
7. The mass G grams of a radioactive sample after time t years is given by the formula 3100 tG e −= .
(a) What is the initial mass of radioactive substance in the sample? (b) Find the half-life of the radioactive substance.
(a) The initial mass was present when 0t = :
3 0
0
100
100100.
G e
e
− ×=
==
So the initial mass was 100 grams.
(b) The half-life is the time t at which 50G = , so
( )
3
3 12
12
100 50
3 log (converting to log form).0 231 (to 3 d.p.).
t
t
e
e
e
t
t
−
−
=
=
− =
=
So the half-life is .0 231 years, roughly . .0 231 356 84 315× = days.
Note
10log could have been
used instead of loge .
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8. The world population, in billions, t years after 1950 is given by .0 0178.2 54 tP e= .
(a) What was the world population in 1950? (b) Find, to the nearest year, the time taken for the world population to
double.
(a) For 1950, 0t = :
.0 0178 0
0
.2 54
.2 54
.2 54.
P e
e
×=
==
So the world population in 1950 was .2 54 billion.
(b) For the population to double:
.0 0178
.0 0178
. .2 54 2 2 54
2.0 0178 log 2 (converting to log form)
.38 94 (to 2 d.p.).
t
t
e
e
ett
= ×
===
So the population doubled after 39 years (to the nearest year).
6 Graphing with Logarithmic Axes EF It is common in applications to find an exponential relationship between variables; for instance, the relationship between the world population and time in the example above. Given some data (e.g. from an experiment) we would like to find an explicit equation for the relationship.
Relationships of the form y = abx
Suppose we have an exponential graph xy ab= , where , 0a b > .
O
a
y
x
xy ab=
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Taking logarithms we find that
( )log log
log loglog log .
xe e
xe e
e e
y ab
a ba x b
=
= += +
We can scale the y-axis so that logeY y= ; the Y-axis is called a logarithmic axis. Now our relationship is of the form ( )log loge eY b x a= + , which is a straight line in the ( ),x Y -plane.
Since this is just a straight line, we can use known points to find the gradient loge b and the Y-axis intercept loge a . From these we can easily find the values of a and b, and hence specify the equation xy ab= .
EXAMPLES
1. The relationship between two variables, x and y, is of the form xy ab= , where a and b are constants. An experiment to test this relationship produced the data shown in the graph, where loge y is plotted against x.
Find the values of a and b. We ned to obtain a straight line equation:
( )
log log (taking logs of both sides)
log log loglog log log
log log .
x
xe e
xe e e
e e e
e e
y ab
y ab
y a by a x b
Y b x a
=
=
= += +
= +
From the graph, the Y-axis intercept is log 3e a = ; so 3.a e=
O
loge y
x
( )7,5
3
O
Y
x
( )log loge eY b x a= +
loge agradient is loge b
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Using the gradient formula:
27
27
5 3log7 0
.
e b
b e
−=
−=
=
2. The results from an experiment were noted as follows:
The relationship between these data can be written in the form xy ab= .
Find the values of a and b, and state the formula for y in terms of x.
We need to obtain a straight line equation:
( )
log log (taking logs of both sides)
log log loglog log log
log log log .
x
xe e
xe e e
e e e
e e e
y ab
y ab
y a by a x b
y b x a
=
=
= += +
= +
We can find the gradient loge b (and hence b), using two points on the line: using ( ). .1 30,2 04 and ( ). .2 80,3 14 ,
. .3 14 2 04 .log 0 73 (to 2 d.p.).. .2 80 1 30e b −= =
−
So 0.73 .2 08b e= = (to 2 d.p.).
Note that .log 0 73 loge ey x a= + . We can work out loge a (and hence a) by substituting a point into this equation: using ( ). .1 30,2 04 ,
.1 09
. . .2 04 0 73 1 30 log. . .log 2 04 0 73 1 30.1 09 (to 2 d.p.)
.2 97 (to 2 d.p.).
e
e
aa
a e
= × += − ×=
==
Therefore . .2 97 2 08xy = × .
x 1⋅30 2⋅00 2⋅30 2⋅80 2⋅04 2⋅56 2⋅78 3⋅14
Note Depending on the points used, slightly different values for a and b may be obtained.
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Equations in the form y = axb
Another common relationship is by ax= , where 0a > . In this case, the relationship can be represented by a straight line if we change both axes to logarithmic ones.
EXAMPLE
3. The results from an experiment were noted as follows:
The relationship between these data can be written in the form by ax= .
Find the values of a and b, and state the formula for y in terms of x.
We need to obtain a straight line equation:
10 10
10 10 10
10 10 10
10
log log (taking logs of both sides)
log log loglog log log
log .
b
b
b
y ax
y ax
y a xy a b x
Y bX a
=
=
= += += +
We can find the gradient b, using two points on the line: using ( ) 1·70,1·33 and ( ) 2·85, 2·01 ,
2·01 1·33 0·59 (to 2 d.p.)2·85 1·70
b −= =
−
So 10 10 10log 0·59log logy x a= + .
Now we can work out a by substituting a point into this equation: using ( ) 1·70,1·33 ,
10
10
0·33
1·33 0·59 1·70 loglog 1·33 0·59 1·70
0·33
102·14 (to 2 d.p.).
aa
a
= × += − ×=
==
Therefore 0·592·14y x= .
1⋅70 2⋅29 2⋅70 2⋅85 1⋅33 1⋅67 1⋅92 2⋅01
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7 Graph Transformations EF Graph transformations were covered in the notes on Functions and Graphs, but we can now look in more detail at applying transformations to graphs of exponential and logarithmic functions.
EXAMPLES
1. Shown below is the graph of ( )y f x= where ( ) 3logf x x= .
(a) State the value of a. (b) Sketch the graph of ( )2 1y f x= + + .
(a) 3
2
log 9
2 (since 3 9).
a =
= =
(b) The graph shifts two units to the left, and one unit upwards:
2. Shown below is part of the graph of 5logy x= .
Sketch the graph of ( )5
1logy x= .
( )5
15
5
1log
loglog
y x
x
x
−
=
=
= −
So reflect in the x-axis.
x O 1
y ( )5
1log xy =
( )5, 1−
x O 1
y ( )y f x=( )5,1
x O
y ( )2 1y f x= + +
( )7,3
( )1,1−
2y = −
x O 1
y ( )y f x=( ) 9 ,a
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3. The diagram shows the graph of 2xy = .
On separate diagrams, sketch the graphs of:
(a) 2 xy −= ;
(b) 22 xy −= .
(a) Reflect in the y-axis:
(b) 2
2
2
2 2
4 2 .
x
x
x
y −
−
−
=
=
= ×
So scale the graph from (a) by 4 in the y-direction:
O
4
y
22 xy −=
( )1,8−
x
O 1
y
2 xy −=( )1,2−
x
O
1
y
x
2xy =
( )1,2
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Exercise 1 : Exponentials
Exercise 2 : Exponential Growth and Decay
17
Exercise 3 : Logarithms
Exercise 4 : Laws of Logarithms
18
Exercise 5 : Further manipulation with Logarithms
19
Exercise 6 : Logs to base 10 and Natural Logarithms
20
21
Exercise 7 : Logarithmic Equations
22
Exercise 8 : Exponential Growth and Decay
23
Exercise 9 :
24
25
Exercise 10 :
26
Exercise 11 : Graph Transformations
Exercise 12
1. The diagram shows part of the graph
of y = log3 x.
(a) Find the values of a and b.
(b) Sketch the graph of y = log3(x + 1) – 3.
2. The diagram shows part of the graph of
y = log2 x.
(a) Find the value of a.
(b) Sketch the graph of y = log2 x – 4.
(c) Sketch the graph of y = log2 8x.
27
3. The diagram shows part of the graph of
y =log5 x.
(a) Find a and b.
(b) Sketch the graph of y = log5 5x.
(c) Sketch the graph of y = log5 x2
(d) Sketch the graph of y = log5 x
1.
4. The diagram shows part of the graph of
y = log4 x.
(a) Find a.
(b) Sketch the graph of y = log4 4x.
(c) Sketch the graph of y = log4 x3
5. The diagram shows part of the graph of
y = loga x.
(a) Determine the value of a.
(b) Sketch the graph of y = loga 9x2
(c) Sketch the graph of y = loga x
1.
6. The diagram shows the graph of
y = logb (x + a).
Find the values of a and b.
28
7. The diagram shows the graph of
y = logb (x + a).
Find a and b.
8. The diagram shows the graph of
y = alog3 (x – b).
Find a and b.
29
ANSWERS Exercise 1 :
Exercise 2 : Exercise 3 :
30
Exercise 4 : Exercise 5 :
Exercise 6 :
31
Exercise 7 :
Exercise 8 :
32
Exercise 9 :
Exercise 10 :
33
Exercise 11 :
34
Higher : Exponential & Logs Revision 20
08 P
1
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2008
P1
3 5
Ans
2007
P2
4
Ans 12.2
2007
P2
1 1 4
Ans
2006
P1
4
Ans a = 2
36
2006
P2
5
Ans
2005
P1
2 1
Ans
2005
P2
4
Ans
2005
P2
1 4
Ans
2004
P1
4
Ans
37
2004
P2
3 4
Ans
2003
P1
4
Ans
2003
P2
2 3
Ans
2002
W
P1
3
Ans
38
2002
W P
2
1 4
Ans
2002
P1
4
Ans
2002
P2
3
Ans
2001
P1
3
Ans x = 81
2001
P1
1 3
39
Ans
2001
P2
3
Ans
2000
P1
3
Ans 2
2000
P2
2 4
Ans
Spec
imen
2 P
2
4 3
40
Ans
Spec
imen
2 P
2
3
Ans
Spec
imen
1 P
1
3
Ans
41
Spec
imen
1 P
2
3
Ans
42
Higher Mathematics Logs and Exponentials Homework
43
